SolutionHey Everyone,

Please find below the solution of the given problem.

Rewriting the inequality to easily identify the zero pointsWe want the given algebraic expression to be positive.

So, we need to find the range of values of x for the following inequality:

\((x – 5)^8 (x – 2)^3>0\)

After we find the range, we need to choose the negative integers in this range since the question asks for negative integral values that satisfy the above inequality.

Plotting the zero points and drawing the wavy line:

Required Range:

Per the wavy line drawn above, it is clear that 2 < x < 5 and x > 5 render the given expression positive.

However, question asks us to choose only negative integral values for x.

Note that these ranges do NOT contain any negative integers.

Therefore, no negative integral value exists for x such that the given expression can be rendered positive.

Correct Answer: Option E

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